**4. Computational Scheme**

The formulated model is complex and its analytical solution cannot be found. So, we move towards the approximate solutions of the present problem with the use of very accurate approximating technique known as finite di fference method. This method is directly applied to partial di fferential Equations (13)–(17) to convert into algebraic system of equations which is solved by coding on computing tool FORTRAN package. The backward di fference is used along *x*-axis and central di fference along *y*-axis. The discretization procedure is given below:

$$\frac{\partial \mathcal{U}}{\partial X} = \frac{\mathcal{U}\_{i,j} - \mathcal{U}\_{i,j-1}}{\Delta X}. \tag{18}$$

$$\frac{\partial \mathcal{U}}{\partial \mathcal{Y}} = \frac{\mathcal{U}\_{i+1,j} - \mathcal{U}\_{i-1,j}}{2\Delta \mathcal{Y}}.\tag{19}$$

$$\frac{\partial^2 \mathcal{U}}{\partial Y^2} = \frac{\mathcal{U}\_{i+1,j} - 2\mathcal{U}\_{i,j} + \mathcal{U}\_{i-1,j}}{\Delta Y^2}. \tag{20}$$

The insertion of Equations (18)–(20) into Equations (13)–(17) implies:

$$V\_{i,j} = \left(V\_{i-1,j} - X\_i \frac{\Delta Y}{\Delta X} \{\mathcal{U}\_{i,j} - \mathcal{U}\_{i,j-1}\} + \frac{1}{8} Y\_j \{\mathcal{U}\_{i+1,j} - \mathcal{U}\_{i-1,j}\} - \frac{1}{2} \Delta Y \mathcal{U}\_{i,j}\right) - \frac{\cos X\_i}{\sin X\_i} X\_i \mathcal{U}\_{i,j} \tag{21}$$


With boundary conditions:

$$\begin{aligned} \mathcal{U}\_{i,j} = 0 &= V\_{i,j}, & \overline{\theta}\_{i,j} &= 1, & \overline{\varphi}\_{i,j} &= 1 & \text{at} & \text{ } & \text{ $\boldsymbol{Y}$ } = 0\\ \mathcal{U}\_{i,j} &\to 0 & \quad \overline{\varphi}\_{i,j} &\to 0, & \overline{\theta}\_{i,j} &\to 0 & \text{as} & \text{ } & \text{ $\boldsymbol{Y}$ } \to \infty \end{aligned} \tag{25}$$

### **5. Governing Equations for Plume Region**

Considering the diagram of the geometry, we can see that nanofluid enters from the region-II to region-III. For this region, the aforesaid model is altered and a new model for the plume region is formulated by following [2]:

$$\frac{\partial(\sharp\sharp t)}{\partial\mathfrak{k}} + \frac{\partial(\sharp\sharp t)}{\partial\mathfrak{k}} = 0\tag{26}$$

$$\hat{u}\frac{\partial\hat{\boldsymbol{n}}}{\partial\hat{\boldsymbol{x}}} + \hat{w}\frac{\partial\hat{\boldsymbol{n}}}{\partial\hat{\boldsymbol{z}}} = \nu\frac{1}{2}\frac{\partial}{\partial\hat{\boldsymbol{z}}}\Big(\hat{\boldsymbol{z}}\frac{\partial\hat{\boldsymbol{u}}}{\partial\hat{\boldsymbol{z}}}\Big) - g\beta\Big(\hat{\boldsymbol{\Gamma}} - \hat{\boldsymbol{\Gamma}}\_{\text{co}}\Big) - g\beta\mathbf{g}\{\hat{\boldsymbol{C}} - \hat{\boldsymbol{C}}\_{\text{co}}\Big) - \frac{\sigma\_{0}\theta\_{0}^{2}}{\rho}\hat{\boldsymbol{u}}\tag{27}$$

$$\boldsymbol{\hat{u}}\frac{\partial\boldsymbol{\Upsilon}}{\partial\boldsymbol{\xi}} + \boldsymbol{\hat{w}}\frac{\partial\boldsymbol{\Upsilon}}{\partial\boldsymbol{\Xi}} = \boldsymbol{a}\frac{1}{2}\frac{\partial}{\partial\boldsymbol{\Xi}}\Big{(}\boldsymbol{\Xi}\frac{\partial\boldsymbol{\Upsilon}}{\partial\boldsymbol{\Xi}}\Big{)} + \boldsymbol{\pi}\Big{\Big{(}}\boldsymbol{D}\_{\text{B}}\frac{\partial\overline{\mathbf{C}}}{\partial\boldsymbol{\Xi}}\frac{\partial\boldsymbol{\Upsilon}}{\partial\boldsymbol{\Xi}} + \frac{\boldsymbol{D}\_{\text{T}}}{T\_{\text{co}}}\Big{(}\frac{\partial\boldsymbol{\Upsilon}}{\partial\boldsymbol{\Xi}}\Big{)}^{2}\Big{)} + \frac{\boldsymbol{Q}\_{0}}{\mu\mathbf{C}\_{p}}\frac{\mu^{2}}{G\_{r}^{1/2}}\boldsymbol{\theta}\tag{28}$$

$$\hbar \frac{\partial \hat{\mathbf{C}}}{\partial \hat{\mathbf{t}}} + \hbar \frac{\partial \hat{\mathbf{C}}}{\partial \hat{\mathbf{z}}} = D\_B \frac{1}{2} \frac{\partial}{\partial \hat{\mathbf{z}}} \Big( \hat{\mathbf{z}} \frac{\partial \hat{\mathbf{C}}}{\partial \hat{\mathbf{z}}} \Big) + \frac{D\_T}{\hat{\mathbf{T}}\_{\text{co}}} \frac{1}{2} \frac{\partial}{\partial \hat{\mathbf{z}}} \Big( \hat{\mathbf{z}} \frac{\partial \hat{\mathbf{T}}}{\partial \hat{\mathbf{z}}} \Big) \tag{29}$$

With boundary conditions:

$$\begin{aligned} \label{eq:SDAC1} \hat{w} &= \frac{\partial \hat{u}}{\partial \hat{z}} = \frac{\partial \hat{\Upsilon}}{\partial \hat{z}} = 0 \qquad \text{at} \qquad \hat{z} = 0, \\\hat{u} &\to 0 = \hat{w} \quad , \hat{\Upsilon} \to \hat{\Upsilon}\_{\infty} \quad , \quad \hat{\mathscr{C}} \to \hat{\mathscr{C}}\_{\infty} \quad \text{as} \quad \hat{z} \to \infty. \end{aligned} \tag{30}$$

Dimensionless variables:

$$\begin{aligned} \mathbf{x} &= \frac{\mathbf{\hat{x}}}{a} \quad , \quad z = \mathbf{G}\_r^{-1/4} \frac{\mathbf{\hat{z}}}{z} \quad , z = \frac{\mathbf{\hat{z}}}{a} \quad , u = \mathbf{G}\_r^{-1/2} \frac{a}{\nu} \mathbf{\overline{u}} \quad , \quad w = \mathbf{G}\_r^{-1/4} \frac{a}{\nu} \mathbf{\hat{w}}\\ \mathbf{G}\_r &= \frac{\mathbf{g} \mathbf{\hat{z}} \Delta T a^3}{\nu^2} \end{aligned} \qquad , \quad \mathbf{G}\_{\text{rc}} = \frac{\mathbf{g} \mathbf{\hat{z}} \Delta T a^3}{\nu^2} \quad , \quad \tau = \frac{(\rho \mathbf{c})\_p}{(\rho \mathbf{c})\_f} \end{aligned} \tag{31}$$

Dimensionless form of system of equations:

$$\frac{\partial(zu)}{\partial \mathbf{x}} + \frac{\partial(zw)}{\partial \mathbf{z}} = 0 \tag{32}$$

$$u\frac{\partial u}{\partial x} + w\frac{\partial u}{\partial z} = \frac{1}{z}\frac{\partial}{\partial z}\Big(z\frac{\partial u}{\partial z}\Big) - \theta - q - M\tag{33}$$

$$u\frac{\partial\theta}{\partial x} + w\frac{\partial\theta}{\partial z} = \frac{1}{\text{Pr}}\frac{1}{z}\frac{\partial}{\partial z}\Big(z\frac{\partial\theta}{\partial z}\Big) + \text{Nb}\frac{\partial\phi}{\partial z}\frac{\partial\theta}{\partial z} + \text{Nt}\Big(\frac{\partial\theta}{\partial z}\Big)^2 + \text{Q}\theta\tag{34}$$

$$u\frac{\partial\varphi}{\partial x} + w\frac{\partial\varphi}{\partial z} = \frac{1}{\text{Sc}} \left( \frac{1}{z} \frac{\partial}{\partial z} \Big| z \frac{\partial\varphi}{\partial z} \right) + \frac{\text{Nt}}{\text{Nb}} \frac{1}{z} \frac{\partial}{\partial z} \Big( z \frac{\partial\theta}{\partial z} \Big) \tag{35}$$

With boundary conditions:

$$\begin{aligned} w &= \frac{\partial u}{\partial z} = \frac{\partial \theta}{\partial z} = 0 \qquad \text{at} \quad z = 0\\ u &\to 0 \quad \quad \quad \quad \quad \quad \quad \quad \theta \to 0 \qquad \quad \quad \quad \quad \quad z \to \infty \end{aligned} \tag{36}$$

For the convenient form of the integration, we use the following variables for the required form:

$$\begin{aligned} \mu &= \mathbf{x}^{\frac{1}{2}} L(X, Z) \quad , \quad W = \mathbf{x}^{-\frac{1}{4}} W(X, Z) \; , \; Z = \mathbf{x}^{-\frac{1}{4}} Z, \\ \theta &= \overline{\theta}(X, Z) \quad , \quad \varphi = \overline{\varphi}(X, Z) \; , \; \mathbf{x} = X. \end{aligned} \tag{37}$$

Using the above primitive variable formulation, we have the following system of equations:

$$Z\frac{\partial \mathcal{U}}{\partial X} - \frac{Z^2}{4X}\frac{\partial \mathcal{U}}{\partial Z} + \frac{3}{4}Z\mathcal{U} + W + Z\frac{\partial W}{\partial Z} = 0\tag{38}$$

$$X\mathcal{U}\frac{\partial \mathcal{U}}{\partial X} + \frac{1}{2}\mathcal{U}^2 + \left(\mathcal{W} - \frac{1}{4}Z\mathcal{U}\right)\frac{\partial \mathcal{U}}{\partial Z} = \frac{1}{Z}\frac{\partial}{\partial Z}\left(Z\frac{\partial \mathcal{U}}{\partial Z}\right) - \overline{\theta} - \overline{\varphi} - X^{1/2}\mathcal{M}\mathcal{U}.\tag{39}$$

$$X\mathcal{U}\frac{\partial\overline{\partial}}{\partial X} + \left(\mathcal{W} - \frac{1}{4}Z\mathcal{U}\right)\frac{\partial\overline{\partial}}{\partial Z} = \frac{1}{\text{Pr}}\frac{1}{Z}\frac{\partial}{\partial Z}\Big(Z\frac{\partial\overline{\partial}}{\partial Z}\Big) + \mathcal{N}\mathbf{b}\frac{\partial\overline{\partial}}{\partial Z}\frac{\partial\overline{\partial}}{\partial Z} + \mathcal{N}\mathbf{t}\Big(\frac{\partial\overline{\partial}}{\partial Z}\Big)^2 + \mathcal{Q}\overline{\partial}.\tag{40}$$

$$X\mathcal{U}\frac{\partial\overline{\boldsymbol{\varphi}}}{\partial\mathcal{X}} + \left(\mathcal{W} - \frac{1}{4}Z\mathcal{U}\right)\frac{\partial\overline{\boldsymbol{\varphi}}}{\partial\mathcal{Z}} = \frac{1}{\text{Sc}}\Big(\frac{1}{Z}\frac{\partial}{\partial\mathcal{Z}}\Big(Z\frac{\partial\overline{\boldsymbol{\varphi}}}{\partial\mathcal{Z}}\Big) + \frac{\text{Nt}}{\text{Nb}}\frac{1}{Z}\frac{\partial}{\partial\mathcal{Z}}\Big(Z\frac{\partial\overline{\boldsymbol{\theta}}}{\partial\mathcal{Z}}\Big)\Big).\tag{41}$$

With boundary conditions:

$$\mathcal{W} = \frac{\partial \mathcal{U}}{\partial \mathcal{Z}} = \frac{\partial \mathcal{S}}{\partial \mathcal{Z}} = 0 \,, \qquad \overline{\mathcal{q}} = 1 \,, \qquad \overline{\mathcal{\theta}} = 1 \,, \qquad \text{at} \qquad \mathcal{Z} = 0 \,\tag{42}$$

$$\mathcal{U} \to 0 \,, \qquad \overline{\mathcal{q}} \to 0 \,, \qquad \overline{\mathcal{\theta}} \to 0 \, \qquad \text{as} \qquad \mathcal{Z} \to \infty.$$
